Single equations without initial conditions First order equations Equation (1) ode = (x^4-x^3)*u'[x]+2*x^4*u[x] == x^3/3+C 2*x^4*u[x] + (-x^3 + x^4)*Derivative[1][u][x] == C + x^3/3 DSolve[ ode,u,x ] {{u -> Function[x, (6*C - 3*x^2 + 2*x^3)/ (12*(-1 + x)^2*x^2) + E^(-2*x - 2*Log[1 - x])*C[1]]}} Equation (2) ode = -1/2 u'[x] + u[x] == Sin[x] u[x] - Derivative[1][u][x]/2 == Sin[x] DSolve[ ode,u,x ] {{u -> Function[x, E^(2*x)*C[1] + (2*(Cos[x] + 2*Sin[x]))/5]}} Equation (3) ode = y'[x] == y[x]/(y[x] Log[y[x]]+x) Derivative[1][y][x] == y[x]/(x + Log[y[x]]*y[x]) DSolve[ ode,y,x ] DSolve[Derivative[1][y][x] == y[x]/(x + Log[y[x]]*y[x]), y, x] Equation (4) ode = 2 y[x] y'[x]^2 - 2 x y'[x] - y[x] == 0 -y[x] - 2*x*Derivative[1][y][x] + 2*y[x]*Derivative[1][y][x]^2 == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-y[x] - 2*x*Derivative[1][y][x] + 2*y[x]*Derivative[1][y][x]^2 == 0, y, x] Equation (5) ode = y'[x] + y[x] == y[x]^3 Sin[x] y[x] + Derivative[1][y][x] == Sin[x]*y[x]^3 DSolve[ ode,y,x ] {{y -> Function[x, -(E^(2*x)*C[1] + (2*Cos[x])/5 + (4*Sin[x])/5)^(-1/2)]}, {y -> Function[x, (E^(2*x)*C[1] + (2*Cos[x])/5 + (4*Sin[x])/5)^(-1/2)]}, {y -> Function[x, 0]}} Equation (6) ode = y'[x] + P[x] y[x] == Q[x] y[x]^n P[x]*y[x] + Derivative[1][y][x] == Q[x]*y[x]^n The general form of the Bernoulli equation: DSolve[ ode,y,x ] {{y -> Function[x, (E^((-1 + n)*Integrate[P[x], x])* (C[1] + (1 - n)*Integrate[E^ ((1 - n)*Integrate[P[x], x])*Q[x], x]))^ (1 - n)^(-1)]}} Special case: n = 0 DSolve[ ode /. n->0, y,x ] General::intinit: Loading integration packages -- please wait. {{y -> Function[x, C[1]/E^Integrate[P[x], x] + Integrate[E^Integrate[P[DSolve`t], DSolve`t]* Q[DSolve`t], {DSolve`t, 0, x}]/E^Integrate[P[x], x] ]}} Special case: n = 1 DSolve[ ode /. n->1, y,x ] {{y -> Function[x, E^Integrate[-P[x] + Q[x], x]*C[1]]}} DSolve[ ode /. n->Pi, y,x] {{y -> Function[x, (E^((-1 + Pi)*Integrate[P[x], x])* (C[1] + (1 - Pi)* Integrate[E^((1 - Pi)*Integrate[P[x], x])*Q[x], x]))^(1 - Pi)^(-1)]}} And handles n to be complex valued too: DSolve[ ode /. n->I, y,x ] Solve::ifun: Warning: Inverse functions are being used by Solve, so some solutions may not be found. {{y -> Function[x, (E^((-1 + I)*Integrate[P[x], x])* (C[1] + (1 - I)*Integrate[E^ ((1 - I)*Integrate[P[x], x])*Q[x], x]))^ (1/2 + I/2)]}} Equation (7) ode= (x^2-1) y'[x]^2 - 2 x y[x]y'[x] + y^2 - 1 == 0 -1 + y^2 - 2*x*y[x]*Derivative[1][y][x] + (-1 + x^2)*Derivative[1][y][x]^2 == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-1 + y^2 - 2*x*y[x]*Derivative[1][y][x] + (-1 + x^2)*Derivative[1][y][x]^2 == 0, y, x] Equation (8) ode = f[x y'[x]] == g[y'[x]] f[x*Derivative[1][y][x]] == g[Derivative[1][y][x]] DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[f[x*Derivative[1][y][x]] == g[Derivative[1][y][x]], y, x] Equation (9) ode = y'[x] == (3 x^2 - y[x]^2 - 7)/(Exp[y[x]]+2 x y[x]+1) Derivative[1][y][x] == (-7 + 3*x^2 - y[x]^2)/ (1 + E^y[x] + 2*x*y[x]) DSolve[ ode,y,x ] DSolve[Derivative[1][y][x] == (-7 + 3*x^2 - y[x]^2)/(1 + E^y[x] + 2*x*y[x]), y, x] Equation (10) ode = y'[x] == (2 x^3 y[x] - y[x]^4) / (x^4 - 2 x y[x]^3) Derivative[1][y][x] == (2*x^3*y[x] - y[x]^4)/ (x^4 - 2*x*y[x]^3) DSolve[ ode,y,x ] DSolve[Derivative[1][y][x] == (2*x^3*y[x] - y[x]^4)/(x^4 - 2*x*y[x]^3), y, x] Equation (11) ode = y'[x] (y'[x] + y[x]) == x (x + y[x]) Derivative[1][y][x]*(y[x] + Derivative[1][y][x]) == x*(x + y[x]) DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[Derivative[1][y][x]*(y[x] + Derivative[1][y][x]) == x*(x + y[x]), y, x] Factored form of ode11: odeFactored = (y'[x] + y[x] + x)(y'[x] - x) == 0 (-x + Derivative[1][y][x])* (x + y[x] + Derivative[1][y][x]) == 0 DSolve[ odeFactored,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[(-x + Derivative[1][y][x])* (x + y[x] + Derivative[1][y][x]) == 0, y, x] Equation (12) ode = y'[x] == x/(x^2 y[x]^2 + y[x]^5) Derivative[1][y][x] == x/(x^2*y[x]^2 + y[x]^5) DSolve[ ode,y,x ] DSolve[Derivative[1][y][x] == x/(x^2*y[x]^2 + y[x]^5), y, x] Equation (13) ode = y[x] == 2 x y'[x] - a y'[x]^3 y[x] == 2*x*Derivative[1][y][x] - a*Derivative[1][y][x]^3 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[y[x] == 2*x*Derivative[1][y][x] - a*Derivative[1][y][x]^3, y, x] Equation (14) ode = y[x] == 2 x y'[x] - y'[x]^2 y[x] == 2*x*Derivative[1][y][x] - Derivative[1][y][x]^2 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[y[x] == 2*x*Derivative[1][y][x] - Derivative[1][y][x]^2, y, x] Equation (15) ode = y'[x] == Exp[x] y[x]^2 - y[x] + Exp[-x] Derivative[1][y][x] == E^(-x) - y[x] + E^x*y[x]^2 DSolve[ ode,y,x ] DSolve[Derivative[1][y][x] == E^(-x) - y[x] + E^x*y[x]^2, y, x] Equation (16) ode = y'[x] == y[x]^2 - x y[x] + 1 Derivative[1][y][x] == 1 - x*y[x] + y[x]^2 DSolve[ ode,y,x ] DSolve[Derivative[1][y][x] == 1 - x*y[x] + y[x]^2, y, x] Equation (17) ode = y'[x] == (9 x^8 + 1)/(y[x]^2+1) Derivative[1][y][x] == (1 + 9*x^8)/(1 + y[x]^2) DSolve[ ode,y,x ] {{y -> Function[x, (-3*2^(1/3))/ (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3) + (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3)/ (3*2^(1/3))]}, {y -> Function[x, (3*(1 + I*3^(1/2)))/ (2^(2/3)*(81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3)) \ - ((1 - I*3^(1/2))* (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3))/ (6*2^(1/3))]}, {y -> Function[x, (3*(1 - I*3^(1/2)))/ (2^(2/3)*(81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3)) \ - ((1 + I*3^(1/2))* (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3))/ (6*2^(1/3))]}} Equation (18) ode = y[x] == 2 x y'[x] + y[x] y'[x]^2 y[x] == 2*x*Derivative[1][y][x] + y[x]*Derivative[1][y][x]^2 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[y[x] == 2*x*Derivative[1][y][x] + y[x]*Derivative[1][y][x]^2, y, x] Equation (19) ode = y[x] y'[x] - x y'[x]^2 == x y[x]*Derivative[1][y][x] - x*Derivative[1][y][x]^2 == x DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[y[x]*Derivative[1][y][x] - x*Derivative[1][y][x]^2 == x, y, x] Second order equations Equation (20) ode = y''[x] (a x + b)^2 + 4 y'[x] (a x + b) a + 2 y[x] a^2 == 0 2*a^2*y[x] + 4*a*(b + a*x)*Derivative[1][y][x] + (b + a*x)^2*Derivative[2][y][x] == 0 DSolve[ ode,y,x ] {{y -> Function[x, (x*C[1] - C[2])/(b + a*x)^2]}} Equation (21) ode = (x^2-x) u''[x] + (2 x^2+4 x -3) u'[x] + 8 x u[x] == 1 8*x*u[x] + (-3 + 4*x + 2*x^2)*Derivative[1][u][x] + (-x + x^2)*Derivative[2][u][x] == 1 DSolve[ ode,u,x ] DSolve[8*x*u[x] + (-3 + 4*x + 2*x^2)*Derivative[1][u][x] + (-x + x^2)*Derivative[2][u][x] == 1, u, x] Equation (22) ode = (x^2-x) w''[x] + (1-2 x^2) w'[x] + (4 x-2) w[x] == 0 (-2 + 4*x)*w[x] + (1 - 2*x^2)*Derivative[1][w][x] + (-x + x^2)*Derivative[2][w][x] == 0 DSolve[ ode,w,x ] DSolve[(-2 + 4*x)*w[x] + (1 - 2*x^2)*Derivative[1][w][x] + (-x + x^2)*Derivative[2][w][x] == 0, w, x] Equation (23) ode = y''[x] - y'[x] == 2 y[x] y'[x] -Derivative[1][y][x] + Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x] DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-Derivative[1][y][x] + Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x], y, x] Equation (24) ode = y''[x]/y[x] - y'[x]^2/y[x]^2 - 1 + 1/y[x]^3 == 0 -1 + y[x]^(-3) - Derivative[1][y][x]^2/y[x]^2 + Derivative[2][y][x]/y[x] == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-1 + y[x]^(-3) - Derivative[1][y][x]^2/y[x]^2 + Derivative[2][y][x]/y[x] == 0, y, x] Equation (25) ode = y''[x] + 2 x y'[x] == 2 x 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 2*x DSolve[ ode,y,x ] DSolve[2*x*Derivative[1][y][x] + Derivative[2][y][x] == 2*x, y, x] Equation (26) ode = 2 y[x] y''[x] - y'[x]^2 == 1/3 (y'[x] - x y''[x])^2 -Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == (Derivative[1][y][x] - x*Derivative[2][y][x])^2/3 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == (Derivative[1][y][x] - x*Derivative[2][y][x])^2/3, y, x] Equation (27) ode = x y''[x] == 2 y[x] y'[x] x*Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x] DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[x*Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x], y, x] Equation (28) ode = (1-x)(y[x] y''[x] - y'[x]^2) + x^2 y[x]^2 == 0 x^2*y[x]^2 + (1 - x)*(-Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x]) == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[x^2*y[x]^2 + (1 - x)* (-Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x]) == 0, y, x] Equation (29) ode = x y[x] y''[x] + x y'[x]^2 + y[x] y'[x] == 0 y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 + x*y[x]*Derivative[2][y][x] == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 + x*y[x]*Derivative[2][y][x] == 0, y, x] Equation (30) ode = y''[x]^2 - 2 y'[x] y''[x] - 2 y[x] y'[x] - y[x]^2 == 0 2 2 Out[1]= -y[x] - 2 y[x] y'[x] - 2 y'[x] y''[x] + y''[x] == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. 2 2 Out[2]= DSolve[-y[x] - 2 y[x] y'[x] - 2 y'[x] y''[x] + y''[x] == 0, y, x] Equation (31) ode = (x^3/2 - x^2) y''[x] + (2 x^2 - 3 x + 1) y'[x] + (x-1) y[x] == 0 (-1 + x)*y[x] + (1 - 3*x + 2*x^2)*Derivative[1][y][x] + (-x^2 + x^3/2)*Derivative[2][y][x] == 0 DSolve[ ode,y,x ] {{y -> Function[x, (C[1] - ((E*Pi)/2)^(1/2)*C[2]* Erfi[(2 - x)^(1/2)/(2^(1/2)*x^(1/2))])/ (E^x^(-1)*(2 - x)^(1/2)*x^(1/2))]}} Equation (32) ode = y''[x] - 2 x y'[x] + 2 y[x] == 3 2*y[x] - 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 3 DSolve[ ode,y,x ] DSolve[2*y[x] - 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 3, y, x] Equation (33) ode = Sqrt[x] y''[x] + 2 x y'[x] + 3 y[x] == 0 3*y[x] + 2*x*Derivative[1][y][x] + x^(1/2)*Derivative[2][y][x] == 0 DSolve[ ode,y,x ] DSolve[3*y[x] + 2*x*Derivative[1][y][x] + x^(1/2)*Derivative[2][y][x] == 0, y, x] Equation (34) ode = x^2 y''[x] + 3 x y'[x] + 2 y[x] == 1/(y[x]^3 x^4) In[2]:= DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. 2 1 Out[2]= DSolve[2 y[x] + 3 x y'[x] + x y''[x] == --------, y, x] 4 3 x y[x] Equation (35) ode = y''[x] - 2/x^2 y[x] == 7 x^4 + 3 x^3 (-2*y[x])/x^2 + Derivative[2][y][x] == 3*x^3 + 7*x^4 DSolve[ ode,y,x ] DSolve[(-2*y[x])/x^2 + Derivative[2][y][x] == 3*x^3 + 7*x^4, y, x] Equation (36) ode = y''[x] + y[x] == Csc[x] y[x] + Derivative[2][y][x] == Csc[x] DSolve[ ode,y[x],x ] {{y[x] -> C[2]*Cos[x] - x*(Cos[x] + I*Sin[x]) - C[1]*Sin[x] + I/2*Log[-1 + Cos[2*x] + I*Sin[2*x]]* (Cos[x] + I*Sin[x])*(-1 + Cos[2*x] - I*Sin[2*x])}} Higher order equations Equation (37) ode = y'''''''[x]-14 y''''''[x] + 80 y'''''[x] - 242 y''''[x] + 419 y'''[x] - 416 y''[x] + 220 y'[x] - 48 y[x] == 0 -48*y[x] + 220*Derivative[1][y][x] - 416*Derivative[2][y][x] + 419*Derivative[3][y][x] - 242*Derivative[4][y][x] + 80*Derivative[5][y][x] - 14*Derivative[6][y][x] + Derivative[7][y][x] == 0 DSolve[ ode,y,x ] DSolve::dsdeg: Warning: Differential equation of order higher than four encountered. DSolve may not be able to find the solution. {{y -> Function[x, E^x*C[1] + E^x*x*C[2] + E^x*x^2*C[3] + E^(2*x)*C[4] + E^(2*x)*x*C[5] + E^(3*x)*C[6] + E^(4*x)*C[7]]}} Equation (38) ode = y''''[x] - 4/x^2 y''[x] + 8/x^3 y'[x] - 8/x^4 y[x] == 0 (-8*y[x])/x^4 + (8*Derivative[1][y][x])/x^3 - (4*Derivative[2][y][x])/x^2 + Derivative[4][y][x] == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[(-8*y[x])/x^4 + (8*Derivative[1][y][x])/x^3 - (4*Derivative[2][y][x])/x^2 + Derivative[4][y][x] == 0, y, x] Equation (39) ode = (1+x+x^2) y'''[x] + (3 + 6 x) y''[x] + 6 y'[x] == 6 x 6*Derivative[1][y][x] + (3 + 6*x)*Derivative[2][y][x] + (1 + x + x^2)*Derivative[3][y][x] == 6*x DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[6*Derivative[1][y][x] + (3 + 6*x)*Derivative[2][y][x] + (1 + x + x^2)*Derivative[3][y][x] == 6*x, y, x] Equation (40) ode = (y'[x]^2+1) y'''[x] - 3 y'[x] y''[x]^2 == 0 -3*Derivative[1][y][x]*Derivative[2][y][x]^2 + (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-3*Derivative[1][y][x]*Derivative[2][y][x]^2 + (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0, y, x] Equation (41) ode = 3 y''[x] y''''[x] - 5 y'''[x]^2 == 0 -5*Derivative[3][y][x]^2 + 3*Derivative[2][y][x]*Derivative[4][y][x] == 0 DSolve[ ode,y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[-5*Derivative[3][y][x]^2 + 3*Derivative[2][y][x]*Derivative[4][y][x] == 0, y, x] Special equations Equation (42) ode = y'[t] + a y[t-1] == 0 a*y[-1 + t] + Derivative[1][y][t] == 0 DSolve[ ode,y,t ] {{y -> Function[t, C[1] - a*Integrate[y[-1 + DSolve`t], {DSolve`t, 0, t}]]}} Equation (43) ode = D[y[x,a],x] == a y[x,a] Derivative[1, 0][y][x, a] == a*y[x, a] DSolve[ ode,y,x ] DSolve::deqx: Supplied equations are not differential equations of the given functions. DSolve[Derivative[1, 0][y][x, a] == a*y[x, a], y, x] Single equations with initial conditions Equation (44) ode = y''''[x] == Sin[x] Derivative[4][y][x] == Sin[x] DSolve[ {ode, y[0]==0,y'[0]==0,y''[0]==0,y'''[0]==0},y,x ] {{y -> Function[x, -x + x^3/6 + Sin[x]]}} Equation (45) ode = x y''[x] + y'[x] + 2 x y[x] == 0 2*x*y[x] + Derivative[1][y][x] + x*Derivative[2][y][x] == 0 DSolve[ {ode, y[0]==1,y'[0]==0},y,x ] Infinity::indet: Indeterminate expression ComplexInfinity + <<1>> encountered. Solve::svars: Warning: Equations may not give solutions for all "solve" variables. {{y -> Function[x, BesselY[0, 2^(1/2)*x]*C[2] + BesselJ[0, 2^(1/2)*x]*(1 + C[2]*DirectedInfinity[1])] }} Equation (46) ode = x y'[x]^2 - y[x]^2 + 1 == 0 1 - y[x]^2 + x*Derivative[1][y][x]^2 == 0 DSolve[ {ode, y[0]==1},y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[{1 - y[x]^2 + x*Derivative[1][y][x]^2 == 0, y[0] == 1}, y, x] Equation (47) ode = y''[x] + y[x] y'[x]^3 == 0 y[x]*Derivative[1][y][x]^3 + Derivative[2][y][x] == 0 DSolve[ {ode, y[0]==0, y'[0]==2},y,x ] DSolve::dnim: Built-in procedures cannot solve this differential equation. DSolve[{y[x]*Derivative[1][y][x]^3 + Derivative[2][y][x] == 0, y[0] == 0, Derivative[1][y][0] == 2}, y, x] Systems of equations Equation (48) DSolve[ {x'[t] == - 3 y[t] z[t], y'[t] == 3 x[t] z[t], z'[t] == - x[t] y[t]}, {x,y,z},t ] DSolve[{Derivative[1][x][t] == 3*y[t]*z[t], Derivative[1][y][t] == 3*x[t]*z[t], Derivative[1][z][t] == -(x[t]*y[t])}, {x, y, z}, t] Equation (49) DSolve[ {x'[t] == a[t](y[t]^2-x[t]^2) + 2 b[t] x[t] y[t] + 2 c x[t], y'[t] == b[t](y[t]^2-x[t]^2) - 2 a[t] x[t] y[t] + 2 c y[t]}, {x,y},t ] DSolve[{Derivative[1][x][t] == 2*c*x[t] + 2*b[t]*x[t]*y[t] + a[t]*(-x[t]^2 + y[t]^2), Derivative[1][y][t] == 2*c*y[t] - 2*a[t]*x[t]*y[t] + b[t]*(-x[t]^2 + y[t]^2)}, {x, y}, t] Equation (50) DSolve[ {x'[t] == x[t] (1+Cos[t]/(2+Sin[t])), y'[t] == x[t] - y[t]}, {x,y},t ] DSolve[{Derivative[1][x][t] == (1 + Cos[t]/(2 + Sin[t]))*x[t], Derivative[1][y][t] == x[t] - y[t]}, {x, y}, t] Equation (51) DSolve[ {x'[t] == 9 x[t] + 2 y[t], y'[t] == x[t] + 8 y[t]}, {x,y}, t ] {{x -> Function[t, (E^(7*t)/3 + (2*E^(10*t))/3)*C[1] + ((-2*E^(7*t))/3 + (2*E^(10*t))/3)*C[2]], y -> Function[t, (-E^(7*t)/3 + E^(10*t)/3)*C[1] + ((2*E^(7*t))/3 + E^(10*t)/3)*C[2]]}} Equation (52) DSolve[ {x'[t] - x[t] + 2 y[t] == 0, x''[t] - 2 y'[t] == 2 t - Cos[2 t]}, {x,y}, t ] Solve::svars: Warning: Equations may not give solutions for all "solve" variables. Solve::svars: Warning: Equations may not give solutions for all "solve" variables. Equation (53) DSolve[ {y1'[x] == -1/(x (x^2+1)) y1[x] + 1/(x^2 (x^2+1)) y2[x] + 1/x, y2'[x] == -x^2/(x^2+1) y1[x] + (2x^2+1)/(x (x^2+1)) y2[x] + 1}, {y1,y2}, x ] DSolve[{Derivative[1][y1][x] == x^(-1) - y1[x]/(x*(1 + x^2)) + y2[x]/(x^2*(1 + x^2)), Derivative[1][y2][x] == 1 - (x^2*y1[x])/(1 + x^2) + ((1 + 2*x^2)*y2[x])/(x*(1 + x^2))}, {y1, y2}, x] a0=104/25*x^10+(274/25-22/15*Sqrt[-222])*x^8+(7754/75-68/15*Sqrt[-222])*x^6+ (11248/75-194/15*Sqrt[-222])*x^4+(29452/75-296/5*Sqrt[-222])*x^2- 10952/5-148/3*Sqrt[-222] a2=x^12+2*x^10+151/3*x^8+296/3*x^6+5920/9*x^4+10952/9*x^2+5476/9 ode=a2*y''[x]-a0*y[x]==0 In[5]:= DSolve[ode,y[x],x] 10952 74 29452 296 I 2 Out[5]= DSolve[-((-(-----) - 148 I Sqrt[--] + (----- - ----- Sqrt[222]) x + 5 3 75 5 11248 194 I 74 4 7754 68 I 74 6 > (----- - ----- Sqrt[--]) x + (---- - ---- Sqrt[--]) x + 75 5 3 75 5 3 10 274 22 I 74 8 104 x > (--- - ---- Sqrt[--]) x + -------) y[x]) + 25 5 3 25 2 4 6 8 5476 10952 x 5920 x 296 x 151 x 10 12 > (---- + -------- + ------- + ------ + ------ + 2 x + x ) y''[x] == 0 9 9 9 3 3 > , y[x], x]